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I'm having some troubles to "suggest" ARIMA models based only in the series plot, autocorrelation and partial autocorrelation function.

In the time series below, the first plot is the serie (no description of time), the second and third are Autocorreation and Partial Autocorrelation. enter image description here

When the series have trend I know that is not stationary and need differencing, but in this case there is no trend. Apparently this series is nonstationary because the mean look bigger at start, but how can I simply by looking at the series identify that it has mean and constant variance?

This is some slides that I'm using Forecasting using R. There is some statement that say "The ACF of stationary data drops to zero relatively quickly", but how fast? In this case the ACF drops fast, but in some point the autocorrelatios become negative significant.

The plot of Partial Autocorrelation Function shows significant values at lag 1, 2 and 12.

I should consider a model with 1 difference?

What I think in general:

  • The mean at first seems to me to be higher than in the other parts of the process, which indicates that the series is not stationary.

  • The behavior of ACF and PACF looks like an Autoregressive Model

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    $\begingroup$ With a time series like this the plot of the original series does not really suggest a change in mean. The series is suppose to have some random variation. The autocorrelation function and the partial acf give a better pictures. The acf shows an exponential decline indicative of a stationary autoregressive process. Taking that into account with the pacf only having the first lag significant and highly so I would choose AR(1). Keep in mind more than one model could fit well to your data. George Box would recommend parsimony meaning given a set of models to pick from choose the simplest. $\endgroup$ – Michael Chernick Dec 31 '16 at 19:09
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    $\begingroup$ Parsimony is a good principle as a guide for picking a model that is likely to be better at forecasting. @Irish Stat should be consulted on this. He has been doing this for a living for several decades beginning with his autobox product. Often he will ask for your data and then work the problem himself and then come back to the site with lts of graphs and a detailed answer. $\endgroup$ – Michael Chernick Dec 31 '16 at 19:14
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    $\begingroup$ @MichaelChernick ... nice words .. tu . OP please post your original data and I will try explain the sequence off steps to form a "useful model" . $\endgroup$ – IrishStat Dec 31 '16 at 19:24
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    $\begingroup$ I would say and maybe IrishStat would agree with me. You don't take seriously estimates that barely exceed the threshold for significance. Since you are looking at so many lags those two case that barely exceed the threshold should be ignored. Also no other pattern is there to suggest a better model. $\endgroup$ – Michael Chernick Dec 31 '16 at 19:46
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    $\begingroup$ @Roland If you are taking a time series course,you shouldn't be relying on us this much. You should be studying from your text and classroom notes and speaking with your professor when you are confused. I think CV should only be used to give you general hints about concepts. The self-help tag only tell us that you are interested in specific questions about a subject you are studying on your own. It is a little misleading to us to show us a series of very specific questions that are all part of a course you are taking. $\endgroup$ – Michael Chernick Dec 31 '16 at 19:53
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An appropriate for an acf as presented could (would) be (1,0,0)(1,0,0)12 with coefficients of .4 and -.3 . There is not enough data to suggest a level shift as the difference between the two means would not be large enough given the standard variability exhibited in the plot . We routinely simulate "difficult data" in order to test or automatic procedures. I used the simulation feature of AUTOBOX with the followenter image description hereing specified model . THe simulated data was then used to compute the acf and the pacf ... enter image description here .It roughly resembles your acf and pacf. If I had more time I could actually iterate from .4 and -.3 to get closer to your acf.

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  • $\begingroup$ Interesting, then it could even be a SARIMA model. $\endgroup$ – user72621 Dec 31 '16 at 20:07
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    $\begingroup$ which is what I specified. (1,0,0)(1,0,0,)12 $\endgroup$ – IrishStat Dec 31 '16 at 20:09
  • $\begingroup$ @IrishStat I regret that we have gone so far with this question. This is very educational for the CV community and the OP. But we are possibly helping him get a better grade in a course. This was specifically a quiz question! $\endgroup$ – Michael Chernick Dec 31 '16 at 20:30
  • $\begingroup$ I am not sure whether I agree with you. Do you not think that the more elaborate model could be overfitting? Have you abandoned the principle of parsimony? $\endgroup$ – Michael Chernick Dec 31 '16 at 20:32
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    $\begingroup$ @Michael Please don't be overly concerned about how people might be using our site. It is not for us to guess the motivation and circumstances behind every question. The main reason we treat "self-study" questions differently is that they are genuinely different from the real world questions we focus on: they typically use artificial and/or unrealistic data, do not describe the data, are imperative in tone, and can be extremely superficial (such as being couched as multiple choice questions). $\endgroup$ – whuber Dec 31 '16 at 22:38

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